Definition: Logarithmically Convex and Concave Functions

Let $I\in\mathbb R$ be a real interval. A function $f:I\to\mathbb R^*_+$ with positive values is called logarithmically convex (or concave) if the composition of the $f$ and the logarithm $(\ln\circ f):I\to\mathbb R$ is convex (or concave).

Using the inverse exponential functions (real case or complex case), this is equivalent to the following: $f$ is convex if for every $x_1,x_2$ and $0 < t < 1$ $$|f(tx_1+(1-t)x_2|\le |f(x)^tf(x_2)^{1-t}|.$$ $f$ is concave if for every $x_1,x_2$ and $0 < t < 1$ $$|f(tx_1+(1-t)x_2|\ge |f(x)^tf(x_2)^{1-t}|.$$

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983